Math - Nash equilibrium

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Part 1

Consider a variant of electoral competition on the line that captures features of the US presidential election. Voters are divided between two states. State 1 has more electoral college votes than does state 2. The winner is the candidate who obtains the most electoral college votes.

Denote by mi the median's favorite position among the citizens of state i, for i = 1, 2; assume that m2 < m1. Each of two candidates chooses a single position. Each citizen votes non-strategically for the candidate whose position is closest to her favorite position and randomize 50-50 when indifferent between candidates. The candidate who wins a majority of the votes in a state obtains all the state's electoral votes; if for some state the candidates obtain the same number of votes, they each obtain half of the electoral college votes of that state.

Find all NE of the strategic game modeled here. What does the set of NE imply about the role of small and large states in elections? In your answer assume that there is an even number of electoral votes in each district and that if candidates tie in the electoral college then each wins the election with probability 1/2.

Part 2

Consider another variant of electoral competition on the line. This time suppose the candidates, like citizens, only care about policy outcomes and not about winning per se. There are two candidates, each who has a favorite position. Suppose that the candidates' utility for an alternative is decreasing in its distance from their ideal policy. Also assume that the favorite policy of candidate 1 (the Democrat) is less than the median
(m) and the favorite policy of candidate 2 (the Republican) is greater than the median. Also assume if the candidates tie then the policy splits the difference between their policy positions, 1/2(x1+x2). Find all of the Nash equilibria for this game.

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